I am two weeks into my first stats course and already I have noticed that, because my class ignores measure theory, the instructors are being sloppy about explaining which kinds of functions are Riemann integrable and which ones require more advanced tools. In general, I feel a lot of handwaving is going on and it seems like this is a generally accepted practice amongst the stat teachers.
For example of (not necessarily Riemann) handwaving,
$$\int e^{-x^2} dx$$ cannot be expressed in terms of elementary functions but the definite integral
$$\int_{-\infty}^{\infty}e^{-x^2} dx$$
can be.
Another example would be the Dirac delta function, which isn't a function but formed through "distributions" or something.
What are some examples of frequently used rules like these whose calculus is actually much trickier than one might expect?
Perhaps you're looking for something like this. In elementary probability/statistics courses one deals with random variables that are either "discrete" or "continuous". The discrete ones have countably many possible values with positive probabilities; the continuous ones have a density (piecewise continuous in all examples, but easily generalized to Lebesgue integrable once you have that concept). However, these (and mixtures of these) do not by any means exhaust all the possibilities. There is also a class of singular continuous distributions.
A fairly natural example of a random variable with a singular continuous distribution is this. Consider a sequence of independent tosses of a fair coin, and let $X_n$ be $1$ if the $n$'th toss results in heads, $0$ if tails (thus $X_n$ are independent identically-distributed Bernoulli-$1/2$ random variables), and let $$X = 2 \sum_{n=1}^\infty 3^{-n} X_n $$ Then $X$ has a singular continuous distribution supported on the Cantor set. It's not discrete, because $\mathbb P(X=a) = 0$ for every single point $a$, but $\mathbb P(X \in E) = 1$ where the Cantor set $E$ has measure $0$.